Need help on probability question

Question 1 - Consider two urns. Urn 1 contains 4 red balls and 2 white balls, and urn 2 contains 3 balls of each color. If 2 balls are drawn from urn 1 without replacement and transferred to urn 2 and then a ball is drawn from urn 2, what is the probability that the ball drawn from urn 2 will be red . Given that the ball drawn from urn 2 was red, what is the conditional probability that (a) 0 (b) 1 (c) 2 red balls are transfered

Question 2- Let urn 1 contains 4 red balls and 2 white balls, and let the urn 2 contain 3 balls of each color. If a ball is drawn at random from urn 1 and transferred to urn 2 and then a ball is drawn at random from urn 2, what is the probability that the second ball drawn will be red''

pawan said:

Question 1 - Consider two urns. Urn 1 contains 4 red balls and 2 white balls, and urn 2 contains 3 balls of each color. If 2 balls are drawn from urn 1 without replacement and transferred to urn 2 and then a ball is drawn from urn 2, what is the probability that the ball drawn from urn 2 will be red .

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Suppose that:
$A$ is the event that two red balls are transferred.
$B$ is the event that two white balls are transferred.
$C$ is the event that one of each color is transferred.
$R$ is the event that the ball drawn from urn 2 is red.
\(\displaystyle \begin{align*} \mathcal{P}(A)&=\dfrac{4}{6}\cdot\dfrac{3}{5}= \dfrac{6}{15}\\\mathcal{P}(B)&= \dfrac{2}{6}\cdot\dfrac{1}{5}=\dfrac{1}{15}\\ \mathcal{P}(C)&=2\cdot\dfrac{4}{6}\cdot\dfrac{2}{5}=\dfrac{8}{15}\end{align*}\)

\(\displaystyle \begin{align*}\mathcal{P}(R)&= \mathcal{P}(A\cap R)+ \mathcal{P}(B\cap R)+ \mathcal{P}(C\cap R)\\ &=\mathcal{P}(R|A) \mathcal{P}(A)+ \mathcal{P}(R|B) \mathcal{P}(B)+ \mathcal{P}(R|C)\mathcal{P}(C)\end{align*}\)